系数的级相同的高阶微分方程解的增长性

In this paper,we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations.It is proved that under some conditions for entire functions F,A_(ji) and polynomials P_j(z),Q_j(z)(j=0,1,…,k-1;i=1,2)with degree n≥1,the equation f~(k)+(A_(k-1,1)(z)e~(p_(k-1)(z))+A_(k-1,2)(z)e~(Q_(k-1(z)))/~f~(k-1)+…+(A_(0,1)(z)e~(P_o(z))+A_(0,2)(z)e~(Q_0(z)))f=F,where k≥2,satisfies the properties:When F ≡0,all the non-zero solutions are of infinite order;when F=0,there exists at most one exceptional solution fo with finite order,and all other solutions satisfy λ(f)=λ(f)=σ(f)=∞.

The Growth of Solutions of Higher Order Differential Equations with Coefficients Having the Same Order

In this paper, we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations. It is proved that under some conditions for entire functions $F,A_{ji}$ and polynomials $P_j(z),Q_j(z)~(j=0,1,\ldots,k-1;i=1,2)$ with degree $n\geq 1$, the equation $f^{(k)}+(A_{k-1,1}(z)e^{P_{k-1}(z)}+A_{k-1,2}(z)e^{Q_{k-1}(z)})f^{(k-1)}+\cdots+(A_{0,1}(z)e^{P_{0}(z)}+A_{0,2}(z)e^{Q_{0}(z)})f= F,$ where $k\geq2$, satisfies the properties: When $F\equiv 0$, all the non-zero solutions are of infinite order; when $F\not\equiv 0$, there exists at most one exceptional solution $f_0$ with finite order, and all other solutions satisfy $\overline{\lambda}(f)=\lambda(f)=\sigma(f)=\infty$.